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The present paper considers a diffusive Nicholson's blowflies model with multiple delays under a Neumann boundary condition. Delay independent conditions are derived for the global attractivity of the trivial equilibrium and the positive equilibrium, respectively. Two open problems concerning the stability of positive equilibrium and the occurrence of Hopf bifurcation are proposed.

Since blowflies are important parasites of the sheep industry in some countries such as Australia, based on the experimental data of Nicholson [

It is impossible that the size of the adult blowflies population is independent of a spatial variable; therefore, Yang and So [

Meanwhile, one can consider a nonlinear equation with several delays because of variability of the generation time; for this purpose, Györi and Ladas [

Luo and Liu [

It is of interest to investigate both several temporal and spatial variations of the blowflies population using mathematical models. Hereby, in this paper, we consider the following system:

Though the global attractivity of the nonnegative equilibria of (

It is not difficult to see that if

The rest of the paper is organized as follows. We give some lemmas and definitions in Section

In this section, we will give some lemmas which can be proved by using the similar methods as those in Yang and So [

(i) The solution

(ii) If

Next, we will introduce the concept of lower-upper solution due to Redlinger [

A lower-upper solution pair for (

The following lemma is a special case of Redlinger [

Let

The following lemma gives us boundedness of the solution

(i) The solution

(ii) There exists a constant

Let

Solving the equation, we have

Taking

By Lemma

Note that

Therefore, the formula (

So we complete Lemma

Assume that

By Lemma

Define

By using the similar methods to prove Lemma

Because of

Therefore, from Definition

By Theorem 1 of Luo and Liu [

Hence, we complete the proof of Theorem

If

Let

the function

There are now two possible cases to consider.

Let

From Lemma

Let

We prove that

By induction and direct computation, we have

Similarly, we have

Define

It follows from (

Therefore, from Definition

Note that

Define

Repeating the above procedure, we have the following relation:

By (

Our main results are also valid when

In this section, we will give some numerical simulations to verify our main results in Section

Different parameters will be used for simulations, and some data come from [

Parameters:

Figure

Parameters:

In Section

It is similar to Theorem 3 in Luo and Liu [

If

Figure

Parameters:

From Figure

Parameters:

From Figure

Parameters:

Under suitable conditions, the systems (

Now, we have not intensively studied these two problems. Because the nonmonotonicity of the nonlinear term in (

Project was supported by Hunan Provincial Natural Science Foundation of China (12jj4012) and Research Project of National University of Defense Technology (JC12-02-01).